7.2 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Graphs | Obsidian::STEM | data_structure::graph |
|
Overview
There are two standard ways of representing graphs in memory: adjacency-list representations and adjacency-matrix representations.
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Basic
Using asymptotic notation, how do the number of edges in a graph relate to the number of vertices?
Back: \lvert E \rvert = O(\lvert V^2 \rvert)
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
For graph G = \langle V, E \rangle, why is \lvert E \rvert = O(\lvert V^2 \rvert)?
Back: Because E is a binary relation on V.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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%%ANKI Basic What are the two standard ways of representing graphs in memory? Back: The adjacency-list and adjacency-matrix representation. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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%%ANKI Basic Which standard graph representation is preferred for sparse graphs? Back: Adjacency-list representations. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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%%ANKI Basic Which standard graph representation is preferred for dense graphs? Back: Adjacency-matrix representations. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
When is a graph G = \langle V, E \rangle considered dense?
Back: When \lvert E \rvert \approx \lvert V \rvert^2.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Adjacency-List
Let G = \langle V, E \rangle be a graph. An adjacency-list representation of G has an array of size \lvert V \rvert. Given v \in V, the index corresponding to v contains a linked list containing all adjacent vertices.
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Basic
Let G = \langle V, E \rangle be a graph. It's adjacency-list representation is an array of what size?
Back: \lvert V \rvert
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
The following is an example of what kind of graph representation?
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Back: An adjacency-list representation.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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%%ANKI Basic Are adjacency-list representations used for directed or undirected graphs? Back: Both. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
Let G = \langle V, E \rangle be a graph. What is the sum of its adjacency-list representation's list lengths?
Back: N/A. This depends on whether G is a directed or undirected graph.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
Let G = \langle V, E \rangle be a digraph. What is the sum of its adjacency-list representation's list lengths?
Back: \lvert E \rvert
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
Let G = \langle V, E \rangle be an undirected graph. What is the sum of its adjacency-list representation's list lengths?
Back: 2\lvert E \rvert
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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%%ANKI Basic Which lemma explains the sum of an undirected graph adjacency-list representation's list lengths? Back: The handshake lemma. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
Let G = \langle V, E \rangle. What is the memory usage of its adjacency-list representation?
Back: \Theta(\lvert V \rvert + \lvert E \rvert)
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Adjacency-Matrix
Let G = \langle V, E \rangle be a graph. An adjacency-matrix representation of G is a \lvert V \rvert \times \lvert V \rvert matrix A = (a_{ij}) such that $a_{ij} = \begin{cases} 1 & \text{if } \langle i, j \rangle \in E \\ 0 & \text{otherwise} \end{cases}$
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Basic
Let G = \langle V, E \rangle be a graph. It's adjacency-matrix representation is a matrix of what dimensions?
Back: \lvert V \rvert \times \lvert V \rvert
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
What values are found in an adjacency-matrix representation of a graph?
Back: 0 and/or 1.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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%%ANKI
Basic
The following is an example of what kind of graph representation?
!
Back: An adjacency-matrix representation.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
%%ANKI Basic Are adjacency-matrix representations used for directed or undirected graphs? Back: Both. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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%%ANKI Basic For what graphs are adjacency-matrix representations symmetric along its diagonal? Back: Undirected graphs. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
END%%
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Basic
Why is the adjacency-matrix representation of undirected graph G = \langle V, E \rangle symmetric along its diagonal?
Back: If \langle i, j \rangle \in E then \langle j, i \rangle \in E.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Basic
Let G = \langle V, E \rangle. What is the memory usage of its adjacency-matrix representation?
Back: \Theta(\lvert V \rvert^2)
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Bibliography
- Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).